Exercise 3 Theorem 4. If the time between indepenmdent events is exponentially dis-tributed with common parameter X, then the number of events in an intervalof fixed length T is Poisson with parameter µ = XT.Proof Let Y, be the time until n arrivals, with density given by Equation(17), and let N be the number of arrivals in the interval [0,T). ThenP(N > n) = P(Yn n) – P(N > n+ 1)(20)HenceP(N >n+ 1) = P(Yn+1 < T) =(21)dx =(22)n!Integrating by parts7n-le-Ar dr =(23)n!(п — 1)!loe-AT+ P(Yn < T)(24)n!Equations (20) and (24) give the claimed expression for P(N = n).Exercise 3. The life of an electronic component has the exponential distri-bution with parameter = 0.0004 hrs-. Assuming that the component isreplaced as soon as it fails, find the probability that the number of componentsneeded to accomplish a year of service is 0,1,2,3,4, and more than four.

Given that the life of an electronic component has the exponential distribution with parameter…

Answered: Exercise 3. The life of an electronic…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.4: Values Of The Trigonometric Functions

Problem 22E

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Exercise 3. The life of an electronic component has the exponential distri- bution with parameter = 0.0004 hrs-1. Assuming that the component is replaced as soon as it fails, find the probability that the number of components needed to accomplish a year of service is 0,1,2,3,4, and more than four.